Equations for Dangerous Detected Failure PFD – Derived

Equations are often used to calculate the average probability of failure on demand (PFDavg) of safety instrumented system subsystems.  These equations are sometimes referred to pejoratively as “simplified equations”, but are in fact analytical solutions (i.e., symbolic solutions to fault propagation models such as Markov models and fault trees) where most software that does PFD calculations without these equations use numerical approximations, that while being more flexible, in concept, can only approximate the analytical solution requiring a sufficient number of numerical approximation steps.

While many are satisfied simply using software, and others are content being given the equations, a more rare group desires to see how the equations were derived.  Recently, I taught the ISA EC54 course and the situation arose where I was required to derive the equations that represent the dangerous detected (DD) portion of the PFD for various architectures – specifically 1oo1, 1oo2, and 2oo3.  Since I went through the effort, I figured that I would post the hand-written derivation.  Enjoy…

PFD DD Term Derivation